Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C27 |
| direct product | C2×C53×#1908 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | #121 |
| commutator | C2×C4 | C23×C4×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| prime radical | C22×#41×C53 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C23×C4×C53 ◁ G | |
1698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 75 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 75 points:
(1,2)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75);(3,4,6,5)(7,8,12,14,11,10,18,19)(9,15,16,22,17,13,20,21);(3,5,6,4)(7,9,12,20,11,17,18,16)(8,13,19,21,10,15,14,22);(7,10)(8,11)(9,15)(12,14)(13,17)(16,21)(18,19)(20,22)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 13990);